This quest is flagged through the main story, so don't worry if you don't get it right away from entering Solstheim.
Shortly after getting, and using, the Bend Will shout on the Wind Stone and returning to Skaal, you will be told of a scholar who knows how to use the Black Books. Travel downwards from Skaal Village, hopefully taking care of the two stones on your way as you do so for future's sake, and arrive at Tel Mithryn. Seek out Neloth, the scholar there, and he will inform you of your destination: Nchardak.
Dealing with the enemies outside, he will unlock the Nchardak Reading Room. Once you enter, however, you see that it requires a bit more work than that to acquire the Black Book found within. He will point you out over to the room on your right from entering, which will lead you further in to Nchardak (Nchardak Great Chamber) and from there, the Nchardak Workshop. Working together with Neloth, you must unlock the protective case of the Black Book, and to do so, you must solve the puzzles of the Dwemer ruins. Thankfully, the quest is rather straight-forward from there.
You can tell what sort of control pedestal it is by the color. If it is red, it is a simple toggle and you will spend none of your cubes to operate it. If it's blue, you must temporarily place a cube there to power up whatever device it is for.
Cube #1: Automatic upon entering the Nchardak Great Chamber and letting Neloth ramble on about the ruins.
Cube #2: Automatic upon entering the Nchardak Workshop, powering a fire spout trap on the walls.
Cube #3: You will find yourself in Nchardak Workshop, in an large room with a bridge stretched across the top and the floor beneath housing tables and, once the room is drained, Dwarven Spheres. Use both cubes on the Control Pedestals at the top, then clear the room on the bottom. Return to the cubes and pick one up, then return back to the bottom. You should see not one, but two more controls on the bottom. One activates the stairs leading up to the cube, while the second removes a golden plate that will bar entry. You may either remove the plate at no expense to your cubes, or simply just walk around, climb up, and claim the cube.
Cube #4: Flood the previous room once more by claiming all of your cubes, and the water will rise slightly higher than before, allowing access to the room south of where you once were. If you tried this room before the third cube was claimed, you'd find that it wasn't filled enough to proceed further. Simply run forwards once you get back on land and you will find the cube powering up the pumps in this room.
Unfortunately, this won't be enough once you return to the Great Chamber. Use two of the Cubes you've claimed to operate the pumps and drain some more of the room, allowing access to the Nchardak Aqueduct.
The puzzle here involves operating the red control pedestals to lower the bridges and proceed in. Operate the left and right pedestals (no particular order) to set the bridges in the right order. This will lead you up to a blue pedestal to drain the rest of the room. Neloth will volunteer himself to stay behind to claim the last, leaving you to brave the swinging blades and traps to claim the last cube.
Swim back to the Great Chamber, remove one of the cubes from the draining systems, and operate the boilers. Return to the Reading Room after that and you, or Neloth, can operate the switch to unlock the book. Once it completely rises, you've completed the quest and can claim your reward of the Black Book: Epistolary Acumen.